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The foundations of the rigorous study of \textit{analysis} were laid in the nineteenth century, notably by the mathematicians Cauchy and Weierstrass. Central to the study of this subject are the formal definitions of \textit{limits} and \textit{continuity}.

Let $D$ be a subset of $\bf R$ and let $f \colon D \to \textbf{R}$ be a real-valued function on $D$. The function $f$ is said to be \textit{continuous} on $D$ if, for all $\epsilon > 0$ and for all $x \in D$, there exists some $\delta > 0$ (which may depend on $x$) such that if $y \in D$ satisfies
\[ |y - x| < \delta \]
then
\[ |f(y) < f(x)| < \epsilon. \]

One may readily verify that if $f$ and $g$ are continuous functions on $D$ then the functions $f + g$, $f - g$ and $f . g$ are continuous. If in addtion $g$ is everywhere non-zero then $f / g$ is continuous.

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